Finite Group Databasebeta

← #13150291

Subgroups — #13150291

Order 1728.

748 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721819C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
4#11721255C212yesmaximal, complemented
5#11721241C212yesmaximal, complemented
6#11720997C212yesmaximal, complemented
7#11720986C212yesmaximal, complemented
8#11720920C212yesmaximal, complemented
9#11720914C212yesmaximal, complemented
10#10594980—31nomaximal
11#10589747—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233C2212yes—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19#88536C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C9×#217C2212yes—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4C2212yes—
24C3×Dic9⋊C4C2212yes—
25C3×Dic9⋊C4C2212yes—
26#88445C2212yes—
27#88445C2212yes—
28#88445C2212yes—
29Dic3⋊Dic9C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32Dic3×Dic9C2212yes—
33#64139—31no—
34C3×#924D311yescharacteristic
35C2×C3×Dic3⋊C4—31no—
36#63966—32no—
37#63952—32no—
38#63897—32no—
39#63888—32no—
40#63848—32no—
41#63845—32no—
42C9×C22⋊Q8—31no—
43#63508—32no—
44#63492—31no—
45#63492—61no—
46C2×Dic9⋊C4D311yescharacteristic, complemented
47#63460—32no—
48#63456—32no—
49#63452—32no—
50#63448—32no—
51#611⋊C4—32no—
52C23×C3×C9C2311yescharacteristic
53C22×C3⋊D9C2312yes—
54C2×C3×C4×C9—41no—
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3×C4×C9C2311yescharacteristic
57#6233—24no—
58C2×C3⋊Dic9C2312yes—
59C4⋊(C3⋊D9)—41no—
60C2×C9×Dic3C2312yes—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9—41no—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C9×Dic6—41no—
67(C3⋊D9)⋊C4—42no—
68#5645—91no—
69#5515—91no—
70C22×C32×C4C2×D311yescharacteristic
71C2×#638—32no—
72C2×C4⋊(C3⋊D3)—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C2×C3×Dic6C2×D311yescharacteristic
76C2×#624—32no—
77#3752—32no—
78C4⋊(C3⋊Dic3)—31no—
79C3×#217C2×D312yes—
80C3×C4⋊Dic3C2×D311yescharacteristic
81C3×Dic3⋊C4—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4C2×D312yes—
84#3724—32no—
85Dic3⋊Dic3—32no—
86Dic3⋊Dic3—32no—
87#3722—32no—
88#3722—32no—
89#3722—32no—
90Dic32—32no—
91C2×Q8×C9—31no—
92C22×C4×C9—21no—
93C22×C4×C9C2×D311yescharacteristic
94C2×#611—32no—
95C2×#611—62no—
96C22×Dic9C2×D311yescharacteristic
97C22×Dic9C2×D311yescharacteristic
98C2×D36—31no—
99C2×D36—61no—
100C2×C4×D9—32no—
101C9×C4⋊C4—32no—
102C9×C4⋊C4—31no—
103C9×C22⋊C4—32no—
104(C2×D9)⋊C4—32no—
105(C2×D9)⋊C4—32no—
106(C2×D9)⋊C4—32no—
107(C2×D9)⋊C4—32no—
108(C2×D9)⋊C4—62no—
109C4⋊Dic9—31no—
110C4⋊Dic9—32no—
111C4⋊Dic9—32no—
112C4⋊Dic9—61no—
113Dic9⋊C4—32no—
114Dic9⋊C4C2×D312yes—
115Dic9⋊C4C2×D312yes—
116C4×Dic9—32no—
117C22×C3×C9—21no—
118C22×C3×C9—21no—
119C22×C3×C9#4112yes—
120C22×C3×C9C2411yescharacteristic
121C2×C3⋊D9—22no—
122C2×C3⋊D9—22no—
123C2×C3⋊D9—22no—
124C3×C4×C9—21no—
125C3×C4×C9—21no—
126C3⋊Dic9—22no—
127C9×Dic3—22no—
128C9×Dic3—22no—
129C3×Dic9—22no—
130C3×Dic9—22no—
131C3×C22⋊Q8—31no—
132C2×C3×C4⋊C4—91no—
133#946—92no—
134#930—91no—
135#930—91no—
136#930—181no—
137#929—92no—
138C4×#67—92no—
139#924D911yescharacteristic, complemented
140C2×Dic3⋊C4—31no—
141#898—92no—
142#898—92no—
143#896—92no—
144#894—92no—
145#894—92no—
146#890—92no—
147#889—92no—
148#886—92no—
149#67⋊C4—92no—
150C23×C32C22×D311yescharacteristic
151C22×C3⋊D3—32no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155#638—64no—
156C2×C3⋊Dic3—32no—
157C4⋊(C3⋊D3)—121no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—121no—
162C2×C3×Dic3C22×D312yes—
163C2×C3×Dic3C22×D312yes—
164C3×Dic6—41no—
165#624—122no—
166C23×C9—21no—
167C23×C9C22×D311yescharacteristic
168C22×D9—32no—
169C22×D9—62no—
170Q8×C9—121no—
171C2×C4×C9—21no—
172C2×C4×C9—21no—
173C2×C4×C9—32no—
174C2×C4×C9—32no—
175C2×C4×C9—41no—
176C2×C4×C9—42no—
177C2×C4×C9C22×D311yescharacteristic
178C2×C4×C9C22×D311yescharacteristic
179#611—64no—
180#611—124no—
181C2×Dic9—22no—
182C2×Dic9—32no—
183C2×Dic9—41no—
184C2×Dic9—62no—
185C2×Dic9C22×D312yes—
186C2×Dic9C22×D312yes—
187D36—121no—
188D36—122no—
189C4×D9—122no—
190#539—271noSylow
191C2×C3×C9—22no—
192C2×C3×C9C2×#4111yescharacteristic
193C2×C3×C9H(5,𝔽2)11yescharacteristic
194C2×C3×C92−1+411yescharacteristic
195C3⋊D9—42no—
196C2×C3×Q8—31no—
197C22×C3×C4—21no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C22×C3×C4C2×D911yescharacteristic
201C22×C3×C4D3211yescharacteristic
202C2×#67—92no—
203C2×#67—92no—
204C2×#67—182no—
205C22×Dic3—31no—
206C22×Dic3—31no—
207C2×D12—91no—
208C2×D12—91no—
209C2×D12—181no—
210C2×C4×D3—92no—
211C2×C4×D3—92no—
212C2×Dic6C2×D911yescharacteristic
213C3×C4⋊C4—32no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C22⋊C4—32no—
218#217C2×D912yes—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—92no—
221(C2×D3)⋊C4—92no—
222(C2×D3)⋊C4—92no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—182no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—181no—
233C4⋊Dic3C2×D911yescharacteristic
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4C2×D912yes—
241C4×Dic3—92no—
242C4×Dic3—92no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32#23912yes—
246C22×C32C23×D311yescharacteristic
247C2×C3⋊D3—62no—
248C2×C3⋊D3—62no—
249C2×C3⋊D3—62no—
250C32×C4—21no—
251C32×C4—21no—
252C3⋊Dic3—62no—
253C3×Dic3—22no—
254C3×Dic3—22no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—22no—
261C22×C9—41no—
262C22×C9—41no—
263C22×C9#23512yes—
264C22×C9C23×D311yescharacteristic
265C2×D9—62no—
266C2×D9—62no—
267C2×D9—62no—
268C2×D9—122no—
269C2×D9—122no—
270C2×D9—122no—
271C4×C9—22no—
272C4×C9—21no—
273C4×C9—21no—
274C4×C9—41no—
275C4×C9—62no—
276C4×C9—62no—
277Dic9—22no—
278Dic9—22no—
279Dic9—62no—
280Dic9—122no—
281#126—272no—
282#125—272no—
283#123—272no—
284#123—272no—
285C22⋊Q8—31no—
286C22⋊Q8—272no—
287#121—271no—
288C4×D4—272no—
289C2×C4⋊C4—91no—
290C3×C9#53911yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3C22×D911yescharacteristic
293C23×C3C2×D3211yescharacteristic
294C22×D3—92no—
295C22×D3—92no—
296C22×D3—182no—
297C3×Q8—121no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—182no—
308C2×C3×C4—361no—
309C2×C3×C4C22×D911yescharacteristic
310C2×C3×C4C22×D911yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313#67—184no—
314#67—184no—
315#67—364no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—62no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—121no—
322C2×Dic3—182no—
323C2×Dic3C22×D912yes—
324C2×Dic3C22×D912yes—
325D12—361no—
326D12—361no—
327D12—362no—
328C4×D3—362no—
329C4×D3—362no—
330Dic6—41no—
331C2×C32—22no—
332C2×C32#100411yescharacteristic
333C2×C32C2×#23911yescharacteristic
334C2×C32#101011yescharacteristic
335C3⋊D3—122no—
336C2×C9—22no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—42no—
341C2×C9C2×#23511yescharacteristic
342C2×C9#100411yescharacteristic
343C2×C9#100711yescharacteristic
344D9—122no—
345D9—242no—
346C2×Q8—31no—
347C2×D4—272no—
348C2×D4—271no—
349C22×C4—91no—
350C22×C4—91no—
351C22×C4—272no—
352C22×C4D3×D911yescharacteristic
353C4⋊C4—32no—
354C4⋊C4—31no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—271no—
361C22⋊C4—32no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C22⋊C4—272no—
365C22⋊C4—272no—
366C42—272no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3#370112yes—
376C22×C3C23×D911yescharacteristic
377C22×C3#379912yes—
378C22×C3C22×D3211yescharacteristic
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—182no—
383C2×D3—182no—
384C2×D3—182no—
385C2×D3—362no—
386C2×D3—362no—
387C2×D3—362no—
388C3×C4—22no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—41no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—182no—
397C3×C4—182no—
398Dic3—22no—
399Dic3—22no—
400Dic3—62no—
401Dic3—62no—
402Dic3—182no—
403Dic3—182no—
404Dic3—362no—
405C32#564511yescharacteristic
406C9—21no—
407C9#551511yescharacteristic, complemented
408C23—272no—
409C23C2×D3×D911yescharacteristic
410Q8—121no—
411D4—544no—
412D4—1081no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—41no—
416C2×C4—92no—
417C2×C4—92no—
418C2×C4—182no—
419C2×C4—272no—
420C2×C4—361no—
421C2×C4—1082no—
422C2×C4C2×D3×D911yescharacteristic
423C2×C4C2×D3×D911yescharacteristic
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—42no—
430C2×C3#6371011yescharacteristic
431C2×C3C2×#370111yescharacteristic
432C2×C3#6371611yescharacteristic
433C2×C3#6429711yescharacteristic
434C2×C3C2×#379911yescharacteristic
435C2×C3#6433011yescharacteristic
436D3—362no—
437D3—362no—
438D3—722no—
439C22—21no—
440C22—21no—
441C22—542no—
442C22—542no—
443C22—542no—
444C22#8863512yes—
445C22C22×D3×D911yescharacteristic
446C4—21no—
447C4—21no—
448C4—62no—
449C4—62no—
450C4—182no—
451C4—182no—
452C4—542no—
453C3—21no—
454C3#1058974711yescharacteristic, complemented
455C3#1059498011yescharacteristic
456C2—22no—
457C2—1082no—
458C2C2×#8863511yescharacteristic
459C2#1172246411yescharacteristic
460C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radical#11721819C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#539
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×C3×Dic9⋊C4C2semidirect
4. #11721255C2semidirect
5. #11721241C2semidirect
6. #11720997C2semidirect
7. #11720986C2semidirect
8. #11720920C2semidirect
9. #11720914C2semidirect
46. C2×Dic9⋊C4D3semidirect
139. #924D9semidirect
290. C3×C9#539semidirect
407. C9#5515semidirect
454. C3#10589747semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117218192
C9×#9242
C2×C3×Dic9⋊C42
#117212552
#117212412
#117209972
#117209862
#117209202
#117209142
#105949803
#105897473